Simplify the following expression and state the condition under which the simplification is valid. $t = \dfrac{-9x^3 - 99x^2 - 162x}{3x^3 - 6x^2 - 24x}$
Explanation: First factor out the greatest common factors in the numerator and in the denominator. $ t = \dfrac {-9x(x^2 + 11x + 18)} {3x(x^2 - 2x - 8)} $ $ t = -\dfrac{9x}{3x} \cdot \dfrac{x^2 + 11x + 18}{x^2 - 2x - 8} $ Simplify: $ t = - 3 \cdot \dfrac{x^2 + 11x + 18}{x^2 - 2x - 8}$ Since we are dividing by $x$ , we must remember that $x \neq 0$ Next factor the numerator and denominator. $ t = - 3 \cdot \dfrac{(x + 2)(x + 9)}{(x + 2)(x - 4)}$ Assuming $x \neq -2$ , we can cancel the $x + 2$ $ t = - 3 \cdot \dfrac{x + 9}{x - 4}$ Therefore: $ t = \dfrac{ -3(x + 9)}{ x - 4 }$, $x \neq -2$, $x \neq 0$